What is the integral of $e^{2 x^{2}}$ $e^(2x^2)$ ?

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What is the integral of $e^{2 x^{2}}$ $e^(2x^2)$ ?

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Answer 1 · 2015-08-03T13:04:14

$\int e^{2 x^{2}} d x$ $int e^(2x^2) dx$ cannot be expressed using elementary functions. You need the imaginary error function, erfi(x).

Explanation:

The imaginary error function is $\frac{2}{\sqrt{π}} \int e^{x^{2}} d x$ $2/sqrtpi int e^(x^2) dx$

$\int e^{2 x^{2}} d x$ $int e^(2x^2) dx$ can be intergrated using substitution $u = \sqrt{2} x$ $u = sqrt2 x$ so $d u = \sqrt{2} d x$ $du = sqrt2 dx$ and we get:

$\int e^{2 x^{2}} d x = \frac{1}{\sqrt{2}} \int e^{u^{2}} d u$ $int e^(2x^2) dx = 1/sqrt2 int e^(u^2) du$

$= \frac{1}{\sqrt{2}} \frac{\sqrt{π}}{2} erfi u + C$ $= 1/sqrt2 sqrtpi/2 "erfi" u +C$

$= \frac{\sqrt{2 π}}{4} erfi (\sqrt{2} x) + C$ $= sqrt(2pi)/4 "erfi"(sqrt2x) +C$

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What is the integral of $e^{2 x^{2}}$ $e^(2x^2)$ ?

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